More precisely, given a bounded sequence of non-negative integers \({a}=(a_i)_{i\geqslant 0}\)  satisfying certain assumptions, \({a}\)  defines a “pre-substitution” on the countable alphabet \(\mathbb {N}_0\simeq \lbrace [i]\rbrace \) ; see Eq. (REF ).
This pre-substitution can be extended to a substitution \(\varrho ^{ }_{{a}}\)  on a compact alphabet \(\mathcal {A}\)  via a suitable embedding of \(\mathbb {N}_0\)  into a shift space; see [1]} for the complete construction. This substitution \(\varrho ^{ }_{{a}}\) 
on a now compact alphabet \(\mathcal {A}\) 
satisfies several properties that hold for primitive substitutions, including existence of letter frequencies and of an associated geometric substitution with inflation factor \(\lambda \) . We refer to [1]} and [3]}, [4]} for more details. We briefly remark that the tilings generated by these substitutions have infinite local complexity (ILC) and they can be seen as one-dimensional versions of the pinwheel tiling; see [5]}, [6]} for some statistical and spectral properties of pinwheel tilings.
